## Course Standards

## General Course Information and Notes

### General Notes

**Access Courses:** Access courses are intended only for students with a significant cognitive disability. Access courses are designed to provide students with access to the general curriculum. Access points reflect increasing levels of complexity and depth of knowledge aligned with grade-level expectations. The access points included in access courses are intentionally designed to foster high expectations for students with significant cognitive disabilities.

Access points in the subject areas of science, social studies, art, dance, physical education, theatre, and health provide tiered access to the general curriculum through three levels of access points (Participatory, Supported, and Independent). Access points in English language arts and mathematics do not contain these tiers, but contain Essential Understandings (or EUs). EUs consist of skills at varying levels of complexity and are a resource when planning for instruction.

**English Language Development ELD Standards Special Notes Section:**

Teachers are required to provide listening, speaking, reading and writing instruction that allows English language learners (ELL) to communicate information, ideas and concepts for academic success in the content area of Mathematics. For the given level of English language proficiency and with visual, graphic, or interactive support, students will interact with grade level words, expressions, sentences and discourse to process or produce language necessary for academic success. The ELD standard should specify a relevant content area concept or topic of study chosen by curriculum developers and teachers which maximizes an ELL’s need for communication and social skills. To access an ELL supporting document which delineates performance definitions and descriptors, please click on the following link: http://www.cpalms.org/uploads/docs/standards/eld/MA.pdf.

For additional information on the development and implementation of the ELD standards, please contact the Bureau of Student Achievement through Language Acquisition at sala@fldoe.org.

### General Information

**Course Number:**7912070

**Course Path:**

**Abbreviated Title:**ACCESS LIB ARTS MATH

**Number of Credits:**Course may be taken for up to two credits

**Course Length:**Multiple (M) - Course length can vary

**Course Attributes:**

- Class Size Core Required

**Course Type:**Core Academic Course

**Course Status:**Course Approved

**Grade Level(s):**9,10,11,12

**Graduation Requirement:**Mathematics

## Educator Certifications

## Student Resources

## Original Student Tutorials

Learn how to solve rational linear and quadratic equations using cross multiplication in this interactive tutorial.

Type: Original Student Tutorial

Learn how to solve and graph compound inequalities and determine if solutions are viable in part 2 of this interactive tutorial series.

Click **HERE** to open Part 1.

Type: Original Student Tutorial

Learn how to write equations in two variables in this interactive tutorial.

Type: Original Student Tutorial

Learn how to solve and graph one variable inequalities, including compound inequalities, in part 1 of this interactive tutorial series.

Click **HERE** to open Part 2.

Type: Original Student Tutorial

Explore the construction processes for constructing an angle bisector, copying an angle and constructing a line parallel to a given line through a point not on the line using a variety of tools in this interactive, retro video game-themed tutorial.

Type: Original Student Tutorial

Practice identifying faulty reasoning in this two-part, interactive, English Language Arts tutorial. You'll learn what some experts say about year-round schools, what research has been conducted about their effectiveness, and how arguments can be made for and against year-round education. Then, you'll read a speech in favor of year-round schools and identify faulty reasoning within the argument, specifically the use of hasty generalizations.

Make sure to complete Part One before Part Two! **Click HERE to launch Part One.**

Type: Original Student Tutorial

Learn to identify faulty reasoning in this two-part interactive English Language Arts tutorial. You'll learn what some experts say about year-round schools, what research has been conducted about their effectiveness, and how arguments can be made for and against year-round education. Then, you'll read a speech in favor of year-round schools and identify faulty reasoning within the argument, specifically the use of hasty generalizations.

Make sure to complete both parts of this series! Click HERE to open Part Two.

Type: Original Student Tutorial

Plan a paddle board expedition by learning how to do basic geometric constructions including copying a segment, constructing a segment bisector, constructing a segment's perpendicular bisector and constructing perpendicular segments using a variety of tools in this interactive tutorial.

Type: Original Student Tutorial

Examine President John F. Kennedy's inaugural address in this interactive tutorial. You will examine Kennedy's argument, main claim, smaller claims, reasons, and evidence.

In Part Four, you'll use what you've learned throughout this series to evaluate Kennedy's overall argument.

Make sure to complete the previous parts of this series before beginning Part 4.

Type: Original Student Tutorial

Learn how to construct an inscribed square in a circle and why certain constructions are used in this interactive tutorial.

Type: Original Student Tutorial

Examine President John F. Kennedy's inaugural address in this interactive tutorial. You will examine Kennedy's argument, main claim, smaller claims, reasons, and evidence. By the end of this four-part series, you should be able to evaluate his overall argument.

In Part Three, you will read more of Kennedy's speech and identify a smaller claim in this section of his speech. You will also evaluate this smaller claim's relevancy to the main claim and evaluate Kennedy's reasons and evidence.

Make sure to complete all four parts of this series!

Type: Original Student Tutorial

Learn how to construct an inscribed regular hexagon and equilateral triangle in a circle in this interactive tutorial.

Type: Original Student Tutorial

Quadratic functions can be used to model real-world phenomena. Key features of quadratic functions such as maximum values and zeros can often reveal important qualities of these phenomena. By the end of this tutorial, you should be able to find the zeros of a quadratic function and interpret their meaning in real-world contexts.

Type: Original Student Tutorial

Learn how to add and subtract polynomials in this online tutorial. You will learn how to combine like terms and then use the distribute property to subtract polynomials.

This is part 2 of a two-part lesson. Click below to open part 1.

Type: Original Student Tutorial

Learn how to identify monomials and polynomials and determine their degree in this interactive tutorial.

This is part 1 in a two-part series. **Click HERE to open Part 2**.

Type: Original Student Tutorial

Want to learn about Amelia Earhart, one of the most famous female aviators of all time? If so, then this interactive tutorial is for YOU! This tutorial is Part Two of a two-part series. In this series, you will study a speech by Amelia Earhart. You will practice identifying the purpose of her speech and practice identifying her use of rhetorical appeals (ethos, logos, pathos, Kairos). You will also evaluate the effectiveness of Earhart's rhetorical choices based on the purpose of her speech.

Please complete Part One before beginning Part Two. Click HERE to view Part One.

Type: Original Student Tutorial

Want to learn about Amelia Earhart, one of the most famous female aviators of all time? If so, then this interactive tutorial is for YOU! This tutorial is Part One of a two-part series. In this series, you will study a speech by Amelia Earhart. You will practice identifying the purpose of her speech and practice identifying her use of rhetorical appeals (ethos, logos, pathos, Kairos). You will also evaluate the effectiveness of Earhart's rhetorical choices based on the purpose of her speech.

Please complete Part Two after completing this tutorial. Click HERE to view Part Two.

Type: Original Student Tutorial

Learn how living organisms can be organized into food webs and how energy is transferred through a food web from producers to consumers to decomposers. This interactive tutorial also includes interactive knowledge checks.

Type: Original Student Tutorial

Learn to complete the square of a quadratic expression and identify the maximum or minimum value of the quadratic function it defines. In this interactive tutorial, you'll also interpret the meaning of the maximum and minimum of a quadratic function in a real world context.

Type: Original Student Tutorial

Learn to graph linear inequalities in two variables to display their solutions as you complete this interactive tutorial.

Type: Original Student Tutorial

Learn to rewrite products involving radicals and rational exponents using properties of exponents in this interactive tutorial.

Type: Original Student Tutorial

Learn to construct the perpendicular bisector of a line segment using a straightedge and compass with this interactive tutorial.

Type: Original Student Tutorial

Compare and contrast mitosis and meiosis in this interactive tutorial. You'll also relate them to the processes of sexual and asexual reproduction and their consequences for genetic variation.

Type: Original Student Tutorial

Learn how to determine the shape of a cross section created by the intersection of a slicing plane with a pyramid or prism. This task is vital to those that work to create three dimensional objects. Whether it is the inventor of a new toy or the architect of your next house, they must be able to convey their design on paper. The drawings they make represent various cross sections of the finished product. Can you visualize the relationships between two-dimensional and three-dimensional objects? Imagine that Ninja Nancy will slice through this pyramid with her sword. What two-dimensional figures will she reveal?

Type: Original Student Tutorial

Learn how to calculate and interpret an average rate of change over a specific interval on a graph.

Type: Original Student Tutorial

Learn how to explain the steps used to solve a simple equation and provide reasons to support those steps with this interactive tutorial.

Type: Original Student Tutorial

Use properties, postulates, and theorems to prove a theorem about a triangle. In this interactive tutorial, you'll also learn how to prove that a line parallel to one side of a triangle divides the other two proportionally.

Type: Original Student Tutorial

Learn to determine the number of possible solutions for a linear equation with this interactive tutorial.

Type: Original Student Tutorial

Explain why the x-coordinate of the point of intersection of two functions is the solution of the equation f(x) = g(x).

Type: Original Student Tutorial

Have you ever ordered a scoop of ice cream in a cone and wondered how much ice cream actually fits inside the cone? By the end of this tutorial, you should be able to answer this question and solve other real-world problems by using the formula for the volume of a cone.

Type: Original Student Tutorial

Explore the relationship between mutations, the cell cycle, and uncontrolled cell growth which may result in cancer with this interactive tutorial.

Type: Original Student Tutorial

Learn to identify basic trends in the evolutionary history of humans, including walking upright, brain size, jaw size, and tool use in "Climbing Around the Hominin Family Tree" online tutorial.

Type: Original Student Tutorial

Write linear inequalities for different money situations in this interactive tutorial.

Type: Original Student Tutorial

## Educational Games

In this challenge game, you will be solving inequalities and working with graphs of inequalities. Use the "Teach Me" button to review content before the challenge. During the challenge you get one free solve and two hints! After the challenge, review the problems as needed. Try again to get all challenge questions right! Question sets vary with each game, so feel free to play the game multiple times as needed! Good luck!

Type: Educational Game

In this timed activity, students solve linear equations (one- and two-step) or quadratic equations of varying difficulty depending on the initial conditions they select. This activity allows students to practice solving equations while the activity records their score, so they can track their progress. This activity includes supplemental materials, including background information about the topics covered, a description of how to use the application, and exploration questions for use with the java applet.

Type: Educational Game

In this activity, two students play a simulated game of Connect Four, but in order to place a piece on the board, they must correctly solve an algebraic equation. This activity allows students to practice solving equations of varying difficulty: one-step, two-step, or quadratic equations and using the distributive property if desired. This activity includes supplemental materials, including background information about the topics covered, a description of how to use the application, and exploration questions for use with the Java applet.

Type: Educational Game

## Educational Software / Tool

This virtual manipulative can be used to demonstrate and explore the effect of translation, rotation, and/or reflection on a variety of plane figures. A series of transformations can be explored to result in a specified final image.

Type: Educational Software / Tool

## Perspectives Video: Experts

Jump to it and learn more about how quadratic equations are used in robot navigation problem solving!

Download the CPALMS Perspectives video student note taking guide.

Type: Perspectives Video: Expert

The tide is high! How can we statistically prove there is a relationship between the tides on the Gulf Coast and in a fresh water spring 20 miles from each other?

Download the CPALMS Perspectives video student note taking guide.

Type: Perspectives Video: Expert

Statistical analysis played an essential role in using microgravity sensors to determine location of caves in Wakulla County.

Download the CPALMS Perspectives video student note taking guide.

Type: Perspectives Video: Expert

It's important to stay inside the lines of your project constraints to finish in time and under budget. This NASA systems engineer explains how constraints can actually promote creativity and help him solve problems!

Download the CPALMS Perspectives video student note taking guide.

Type: Perspectives Video: Expert

## Perspectives Video: Professional/Enthusiasts

Listen in as a computing enthusiast describes how hexadecimal notation is used to express big numbers in just a little space.

Download the CPALMS Perspectives video student note taking guide.

Type: Perspectives Video: Professional/Enthusiast

Get fired up as you learn more about ceramic glaze recipes and mathematical units.

Download the CPALMS Perspectives video student note taking guide.

Type: Perspectives Video: Professional/Enthusiast

Don't be a shrinking violet. Learn how uniform scaling is important for candy production.

Download the CPALMS Perspectives video student note taking guide.

Type: Perspectives Video: Professional/Enthusiast

See and see far into the future of arts and manufacturing as a technician explains computer numerically controlled (CNC) machining bit by bit.

Download the CPALMS Perspectives video student note taking guide.

Type: Perspectives Video: Professional/Enthusiast

You'll need to bring your computer skills and math knowledge to estimate oil volume and rate as it seeps from the ocean floor. Dive in!

Download the CPALMS Perspectives video student note taking guide.

Type: Perspectives Video: Professional/Enthusiast

## Presentation/Slideshow

This lesson teaches students about the history of the Pythagorean theorem, along with proofs and applications. It is geared toward high school Geometry students that have completed a year of Algebra and addresses the following national standards of the National Council of Teachers of Mathematics and the Mid-continent Research for Education and Learning: 1) Analyze characteristics and properties of two- and three-dimensional geometric shapes and develop mathematical arguments about geometric relationships; 2) Use visualization, spatial reasoning, and geometric modeling to solve problems; 3) Understand and apply basic and advanced properties of the concepts of geometry; and 4) Use the Pythagorean theorem and its converse and properties of special right triangles to solve mathematical and real-world problems. The video portion is about thirty minutes, and with breaks could be completed in 50 minutes. (You may consider completing over two classes, particularly if you want to allow more time for activities or do some of the enrichment material). These activities could be done individually, in pairs, or groups. I think 2 or 3 students is optimal. The materials required for the activities include scissors, tape, string and markers.

Type: Presentation/Slideshow

## Problem-Solving Tasks

Students explore the structure of the operation *s*/(v*n*). This question provides students with an opportunity to see expressions as constructed out of a sequence of operations: first taking the square root of *n*, then dividing the result of that operation into *s*.

Type: Problem-Solving Task

The purpose of this task is to allow students to demonstrate an ability to construct boxplots and to use boxplots as the basis for comparing distributions.

Type: Problem-Solving Task

This problem solving task challenges students to answer probability questions about SAT scores, using distribution and mean to solve the problem.

Type: Problem-Solving Task

This problem could be used as an introductory lesson to introduce group comparisons and to engage students in a question they may find amusing and interesting.

Type: Problem-Solving Task

The purpose of this task is to have students complete normal distribution calculations and to use properties of normal distributions to draw conclusions.

Type: Problem-Solving Task

This task requires students to use the normal distribution as a model for a data distribution. Students must use given means and standard deviations to approximate population percentages.

Type: Problem-Solving Task

The task provides a context to calculate discrete probabilities and represent them on a bar graph.

Type: Problem-Solving Task

This task asks students to use similarity to solve a problem in a context that will be familiar to many, though most students are accustomed to using intuition rather than geometric reasoning to set up the shot.

Type: Problem-Solving Task

In this problem, students are given a picture of two triangles that appear to be similar, but whose similarity cannot be proven without further information. Asking students to provide a sequence of similarity transformations that maps one triangle to the other, using the definition of similarity in terms of similarity transformations.

Type: Problem-Solving Task

This task involves a reasonably direct application of similar triangles, coupled with a moderately challenging procedure of constructing a diagram from a verbal description.

Type: Problem-Solving Task

The purpose of this task is to engage students in geometric modeling, and in particular to deduce algebraic relationships between variables stemming from geometric constraints.

Type: Problem-Solving Task

In this example, students are asked to write a function describing the population growth of algae. It is implied that this is exponential growth.

Type: Problem-Solving Task

Using a chart of diameters of different denominations of coins, students are asked to figure out how many coins fit around a central coin.

Type: Problem-Solving Task

This problem asks students to model phenomena on the surface of the earth by examining the visibility of the lamp in a lighthouse from a boat.

Type: Problem-Solving Task

This problem solving task encourages students to explore why solar eclipses are rare by examining the radius of the sun and the furthest distance between the moon and the earth.

Type: Problem-Solving Task

Using a triangle with line through it, students are tasked to show the congruent angles, and conclude if one triangle is similar to the other.

Type: Problem-Solving Task

This provides an opportunity to model a concrete situation with mathematics. Once a representative picture of the situation described in the problem is drawn (the teacher may provide guidance here as necessary), the solution of the task requires an understanding of the definition of the sine function.

Type: Problem-Solving Task

The goal of this task is to model a familiar object, an Olympic track, using geometric shapes. Calculations of perimeters of these shapes explain the staggered start of runners in a 400 meter race.

Type: Problem-Solving Task

In this problem, geometry is applied to a 400 meter track to find the perimeter of the track.

Type: Problem-Solving Task

In this task, a typographic grid system serves as the background for a standard paper clip. A metric measurement scale is drawn across the bottom of the grid and the paper clip extends in both directions slightly beyond the grid. Students are given the approximate length of the paper clip and determine the number of like paper clips made from a given length of wire.

Type: Problem-Solving Task

In this task, students will provide a sketch of a paper ice cream cone wrapper, use the sketch to develop a formula for the surface area of the wrapper, and estimate the maximum number of wrappers that could be cut from a rectangular piece of paper.

Type: Problem-Solving Task

This problem solving task asks students to explain which measurements are needed to estimate the thickness of a soda can.

Type: Problem-Solving Task

This problem solving task challenges students to find the surface area of a soda can, calculate how many cubic centimeters of aluminum it contains, and estimate how thick it is.

Type: Problem-Solving Task

This is a mathematical modeling task aimed at making a reasonable estimate for something which is too large to count accurately, the number of leaves on a tree.

Type: Problem-Solving Task

This is a mathematical modeling task aimed at making a reasonable estimate for something which is too large to count accurately, the number of leaves on a tree.

Type: Problem-Solving Task

This problem solving task challenges students to apply the concepts of mass, volume, and density in the real-world context to find how many cells are in the human body.

Type: Problem-Solving Task

The goal of this task is to use geometry to study the structure of beehives.

Type: Problem-Solving Task

Reflective of the modernness of the technology involved, this is a challenging geometric modeling task in which students discover from scratch the geometric principles underlying the software used by GPS systems.

Type: Problem-Solving Task

This problem solving task gives an interesting context for implementing ideas from geometry and trigonometry.

Type: Problem-Solving Task

This problem solving task uses the tale of Archimedes and the King of Syracuse's crown to determine the volume and mass of gold and silver.

Type: Problem-Solving Task

This problem solving task asks students to find the area of a triangle by using unit squares and line segments.

Type: Problem-Solving Task

The purpose of the task is to analyze a plausible real-life scenario using a geometric model. The task requires knowledge of volume formulas for cylinders and cones, some geometric reasoning involving similar triangles, and pays attention to reasonable approximations and maintaining reasonable levels of accuracy throughout.

Type: Problem-Solving Task

This task provides a concrete geometric setting in which to study rigid transformations of the plane.

Type: Problem-Solving Task

This task provides an opportunity for students to apply triangle congruence theorems in an explicit, interesting context.

Type: Problem-Solving Task

This problem solving task challenges students to inscribe equilateral triangles and regular hexagons on a circle with a compass and straightedge.

Type: Problem-Solving Task

This problem solving task challenges students to construct a perpendicular bisector of a given segment.

Type: Problem-Solving Task

This task asks students to use a straightedge and compass to construct the line across which a triangle is reflected.

Type: Problem-Solving Task

The purpose of this task is to use geometric and algebraic reasoning to model a real-life scenario. In particular, students are in several places (implicitly or explicitly) to reason as to when making approximations is reasonable and when to round, when to use equalities vs. inequalities, and the choice of units to work with (e.g., mm vs. cm).

Type: Problem-Solving Task

This task presents a context that leads students toward discovery of the formula for calculating the volume of a sphere.

Type: Problem-Solving Task

This task is inspired by the derivation of the volume formula for the sphere. If a sphere of radius 1 is enclosed in a cylinder of radius 1 and height 2, then the volume not occupied by the sphere is equal to the volume of a "double-naped cone" with vertex at the center of the sphere and bases equal to the bases of the cylinder

Type: Problem-Solving Task

This problem solving task challenges students to bisect a given angle.

Type: Problem-Solving Task

This problem solving task challenges students to place a warehouse (point) an equal distance from three roads (lines).

Type: Problem-Solving Task

This problem solving task challenges students to find the perpendicular meeting point of a segment from the center of a circle and a tangent.

Type: Problem-Solving Task

This task provides a construction of the angle bisector of an angle by reducing it to the bisection of an angle to finding the midpoint of a line segment. It is worth observing the symmetry -- for both finding midpoints and bisecting angles, the goal is to cut an object into two equal parts.

Type: Problem-Solving Task

In this task, students use trigonometric functions to model the movement of a point around a wheel and, through space. Students also interpret features of graphs in terms of the given real-world context.

Type: Problem-Solving Task

This problem solving task challenges students to find the linear, exponential and quadratic functions based on two points.

Type: Problem-Solving Task

This problem solving task asks students to predict and model US population based on a chart of US population data from 1982 to 1988.

Type: Problem-Solving Task

This problem solving task asks students to solve five exponential and linear function problems based on a US population chart for the years 1790-1860.

Type: Problem-Solving Task

This simple conceptual problem does not require algebraic manipulation, but requires students to articulate the reasoning behind each statement.

Type: Problem-Solving Task

The purpose of this task is to give students an opportunity to explore various aspects of exponential models (e.g., distinguishing between constant absolute growth and constant relative growth, solving equations using logarithms, applying compound interest formulas) in the context of a real world problem with ties to developing financial literacy skills. In particular, students are introduced to the idea of inflation of prices of a single commodity, and are given a very brief introduction to the notion of the Consumer Price Index for measuring inflation of a body of goods.

Type: Problem-Solving Task

This task gives a variation of real-life contexts which could be modeled by a linear or exponential function. The key distinguishing feature between the two is whether the change by equal factors over equal intervals (exponential functions), or by a constant increase per unit interval (linear functions). The task could either be used as an assessment problem on this distinction, or used as an introduction to the differences between these very important classes of functions.

Type: Problem-Solving Task

This task requires students to use the fact that on the graph of the linear function h(x) = ax + b, the y-coordinate increases by a when x increases by one. Specific values for a and b were left out intentionally to encourage students to use the above fact as opposed to computing the point of intersection, (p,q), and then computing respective function values to answer the question.

Type: Problem-Solving Task

The typical system of equations or inequalities problem gives the system and asks for the graph of the solution. This task turns the problem around. It gives a solution set and asks for the system that corresponds to it. The purpose of this task is to give students a chance to go beyond the typical problem and make the connections between points in the coordinate plane and solutions to inequalities and equations. Students have to focus on what the graph is showing. When you are describing a region, why does the inequality have to go one way or another? When you pick a point that clearly lies in a region, what has to be true about its coordinates so that it satisfies the associated system of inequalities?

Type: Problem-Solving Task

In order to engage this task meaningfully, students must be aware of the convention that va for a positive number a refers to the positive square root of a. The purpose of the task is to show students a situation where squaring both sides of an equation can result in an equation with more solutions than the original one.

Type: Problem-Solving Task

This task is an example of a mathematical modeling problem (SMP 4) and it also illustrates SMP 1 (Making sense of a problem). Students are only told that there are two ingredients in the pasta and they have a picture of the box. It might even be better to just show the picture of the box, or to bring in the box and ask the students to pose the question themselves. The brand of pasta is quite commonly available at supermarkets or health food stores such as Whole Foods and even at Amazon.com. The box has the nutritional label and a reference to the website where the students can find other information about the ingredients

Type: Problem-Solving Task

This task has some aspects of a mathematical modeling problem (SMP 4) and it also illustrates SMP 1 (Making sense of a problem). Students are given all the relevant information on the nutritional labels, but they have to figure out how to use this information. They have to come up with the idea that they can set up two equations in two unknowns to solve the problem.

Type: Problem-Solving Task

This task addresses A-REI.3.6, solving systems of linear equations exactly, and provides a simple example of a system with three equations and three unknown. Two (of many) methods for solving the system are presented. The first takes the given information to make three equations in three unknowns which can then be solved via algebraic manipulation to find the three numbers. The second solution is more clever, creating a single equation in three unknowns from the given information. This equation is then combined with the given information about the sums of pairs of numbers to deduce what the third number is. In reality, this solution is not simpler than the first: rather it sets up a slightly different set of equations which can be readily solved (the key being to take the sum of the three equations in the first solution). It provides a good opportunity for the instructor to show different methods for solving the same system of linear equations.

Type: Problem-Solving Task

The purpose of this task is to continue a crucial strand of algebraic reasoning begun at the middle school level (e.g, 6.EE.5). By asking students to reason about solutions without explicitly solving them, we get at the heart of understanding what an equation is and what it means for a number to be a solution to an equation. The equations are intentionally very simple; the point of the task is not to test technique in solving equations, but to encourage students to reason about them.

Type: Problem-Solving Task

This task gives students an opportunity to work with exponential functions in a real world context involving continuously compounded interest. They will study how the base of the exponential function impacts its growth rate and use logarithms to solve exponential equations.

Type: Problem-Solving Task

This exploratory task requires the student to use properties of exponential functions in order to estimate how much Carbon 14 remains in a preserved plant after different amounts of time.

Type: Problem-Solving Task

This task involves a fairly straightforward decaying exponential. Filling out the table and developing the general formula is complicated only by the need to work with a fraction that requires decisions about rounding and precision.

Type: Problem-Solving Task

In this task students use verbal descriptions to construct and compare linear and exponential functions and to find where the two functions intersect (F-LE.2, F-LE.3, A-REI.11).

Type: Problem-Solving Task

This task is the last in a series of three tasks that use inequalities in the same context at increasing complexity in 6th grade, 7th grade and in HS algebra. Students write and solve inequalities, and represent the solutions graphically.

Type: Problem-Solving Task

The given solutions for this task involve the creation and solving of a system of two equations and two unknowns, with the caveat that the context of the problem implies that we are interested only in non-negative integer solutions. Indeed, in the first solution, we must also restrict our attention to the case that one of the variables is further even. This aspect of the task is illustrative of mathematical practice standard MP4 (Model with mathematics), and crucial as the system has an integer solution for both situations, that is, whether or not we include the dollar on the floor in the cash box or not.

Type: Problem-Solving Task

This task is a somewhat more complicated version of "Accurately weighing pennies I'' as a third equation is needed in order to solve part (a) explicitly. Instead, students have to combine the algebraic techniques with some additional problem-solving (numerical reasoning, informed guess-and-check, etc.) Part (b) is new to this task, as with only two types of pennies the weight of the collection determines how many pennies of each type are in the collection. This is no longer the case with three different weights but in this particular case, a collection of 50 is too small to show any ambiguity. This is part of the reason for part (c) of the question where the weight alone no longer determines which type of pennies are in the roll. This shows how important levels of accuracy in measurement are as the answer to part (b) could be different if we were to measure on a scale which is only accurate to the nearest tenth of a gram instead of to the nearest hundredth of a gram.

Type: Problem-Solving Task

This classroom task is meant to elicit a variety of different methods of solving a quadratic equation (A-REI.4). Some are straightforward (for example, expanding the square on the right and rearranging the equation so that we can use the quadratic formula); some are simple but clever (reasoning from the fact that x and (2x - 9) have the same square); some use tools (using a graphing calculator to graph the functions f(x) = x^2 and g(x) = (2x-90)^2 and looking for values of x at which the two functions intersect). Some solution methods will work on an arbitrary quadratic equation, while others (such as the last three) may have difficulty or fail if the quadratic equation is not given in a particular form, or if the solutions are not rational numbers.

Type: Problem-Solving Task

This problem involves solving a system of algebraic equations from a context: depending how the problem is interpreted, there may be one equation or two. The main work in parts (a) and (b) is in setting up the equation(s) appropriately. Question (c) is more subtle and it requires thinking carefully about the accuracy available in a particular measurement (weight). The first two parts of this task could be used for instructional or assessment purposes while the third part should strictly be implemented for instructional purposes.

Type: Problem-Solving Task

The purpose of this task is to provide an opportunity for students to reason about equivalence of equations. The instruction to give reasons that do not depend on solving the equation is intended to focus attention on the transformation of equations as a deductive step.

Type: Problem-Solving Task

This task requires students to use the fact that the value of an exponential function f(x) = a · b^x increases by a multiplicative factor of b when x increases by one. It intentionally omits specific values for c and d in order to encourage students to use this fact instead of computing the point of intersection, (p,q), and then computing function values to answer the question.

Type: Problem-Solving Task

This problem assumes that students are familiar with the notation x_{0} and Δx. However, the language "successive quotient" may be new.

Type: Problem-Solving Task

This task assumes that students are familiar with the ?x and ?y notations. Students most likely developed this familiarity in their work with slope.

Type: Problem-Solving Task

An important property of linear functions is that they grow by equal differences over equal intervals. In F.LE Equal Differences over Equal Intervals 1, students prove this for equal intervals of length one unit, and note that in this case the equal differences have the same value as the slope.

Type: Problem-Solving Task

This problem-solving task asks students to examine if linear modeling would be appropriate to describe and predict population growth from select years.

Type: Problem-Solving Task

This problem-solving task provides students an opportunity to experiment with modeling real data by using population growth rates from the past two centuries.

Type: Problem-Solving Task

This problem-solving task challenges students to write expressions and create a table to calculate how much money can be gained after investing at different banks with different interest rates.

Type: Problem-Solving Task

This problem-solving task asks students to describe exponential growth through a real-world problem involving the illegal introduction of fish into a lake.

Type: Problem-Solving Task

This problem-solving emphasizes the expectation that students know linear functions grow by constant differences over equal intervals and exponential functions grow by constant factors over equal intervals.

Type: Problem-Solving Task

This is a simple task touching on two key points of functions. First, there is the idea that not all functions have real numbers as domain and range values. Second, the task addresses the issue of when a function admits an inverse, and the process of "restricting the domain" in order to achieve an invertible function.

Type: Problem-Solving Task

The purpose of this task is to give students practice interpreting statements using function notation. It can be used as a diagnostic if students seem to be having trouble with function notation, for example mistakenly interpreting f(x) as the product of f and x.

Type: Problem-Solving Task

This task is meant to be a straight-forward assessment task of graph reading and interpreting skills. This task helps reinforce the idea that when a variable represents time, t = 0 is chosen as an arbitrary point in time and positive times are interpreted as times that happen after that.

Type: Problem-Solving Task

This task addresses a common misconception about function notation.

Type: Problem-Solving Task

This task could be used for assessment or for practice. It allows students to compare characteristics of two quadratic functions that are each represented differently, one as the graph of a quadratic function and one written out algebraically. Specifically, students are asked to determine which function has the greatest maximum and the greatest non-negative root.

Type: Problem-Solving Task

This task requires interpreting a function in a non-standard context. While the domain and range of this function are both numbers, the way in which the function is determined is not via a formula but by a (pre-determined) sequence of coin flips. In addition, the task provides an opportunity to compute some probabilities in a discrete situation. The task could be used to segue the discussion from functions to probability, in particular the early standards in the S-CP domain.

Type: Problem-Solving Task

The purpose of this task is to investigate the meaning of the definition of function in a real-world context where the question of whether there is more than one output for a given input arises naturally. In more advanced courses this task could be used to investigate the question of whether a function has an inverse.

Type: Problem-Solving Task

The purpose of this task to help students think about an expression for a function as built up out of simple operations on the variable and understand the domain in terms of values for which each operation is invalid (e.g., dividing by zero or taking the square root of a negative number).

Type: Problem-Solving Task

This simple task assesses whether students can interpret function notation. The four parts of the task provide a logical progression of exercises for advancing understanding of function notation and how to interpret it in terms of a given context.

Type: Problem-Solving Task

This task asks students to find the average, write an equation, find the domain, and create a graph of the cost of producing DVDs.

Type: Problem-Solving Task

The principal purpose of the task is to explore a real-world application problem with algebra, working with units and maintaining reasonable levels of accuracy throughout. Students are asked to determine which product will be the most economical to meet the requirements given in the problem.

Type: Problem-Solving Task

This task asks students to consider functions in regard to temperatures in a high school gym.

Type: Problem-Solving Task

The purpose of this task is to introduce or reinforce the concept of a function, especially in a context where the function is not given by an explicit algebraic representation. Further, the last part of the task emphasizes the significance of one variable being a function of another variable in an immediately relevant real-life context.

Type: Problem-Solving Task

In this task students are given graphs of quantities related to weather. The purpose of the task is to show that graphs are more than a collection of coordinate points; they can tell a story about the variables that are involved, and together they can paint a very complete picture of a situation, in this case the weather. Features in one graph, like maximum and minimum points, correspond to features in another graph. For example, on a rainy day, the solar radiation is very low, and the cumulative rainfall graph is increasing with a large slope.

Type: Problem-Solving Task

These problems form a bridge between work on functions and work on probability. The task is better suited for instruction than for assessment as it provides students with a non-standard setting in which to interpret the meaning of functions. Students should carry out the process of flipping a coin and modeling this Random Walk in order to develop a sense of the process before analyzing it mathematically.

Type: Problem-Solving Task

This task is designed to get at a common student confusion between the independent and dependent variables. This confusion often arises in situations like (b), where students are asked to solve an equation involving a function, and confuse that operation with evaluating the function.

Type: Problem-Solving Task

This tasks asks students to use functions to predict the price of a pizza on a specific day and find which day the pizza would be cheapest according to a promotion.

Type: Problem-Solving Task

This problem is a simple de-contextualized version of F-IF Your Father and F-IF Parking Lot. It also provides a natural context where the absolute value function arises as, in part (b), solving for x in terms of y means taking the square root of x^2 which is |x|.This task assumes students have an understanding of the relationship between functions and equations.

Type: Problem-Solving Task

This deceptively simple task asks students to find the domain and range of a function from a given context. The function is linear and if simply looked at from a formulaic point of view, students might find the formula for the line and say that the domain and range are all real numbers.

Type: Problem-Solving Task

This problem introduces a logistic growth model in the concrete settings of estimating the population of the U.S. The model gives a surprisingly accurate estimate and this should be contrasted with linear and exponential models.

Type: Problem-Solving Task

This task is for instructional purposes only and students should already be familiar with some specific examples of logistic growth functions. The goal of this task is to have students appreciate how different constants influence the shape of a graph.

Type: Problem-Solving Task

This task can be used as a quick assessment to see if students can make sense of a graph in the context of a real world situation. Students also have to pay attention to the scale on the vertical axis to find the correct match. The first and third graphs look very similar at first glance, but the function values are very different since the scales on the vertical axes are very different. The task could also be used to generate a group discussion on interpreting functions given by graphs.

Type: Problem-Solving Task

In this task, students will use inverse operations to solve the equations for the unknown variable or for the designated variable if there is more than one.

Type: Problem-Solving Task

The purpose of this task is to give students practice writing a constraint equation for a given context. Instruction accompanying this task should introduce the notion of a constraint equation as an equation governing the possible values of the variables in question (i.e., "constraining" said values). In particular, it is worth differentiating the role of constraint equations from more functional equations, e.g., formulas to convert from degrees Celsius to degree Fahrenheit. The task has students interpret the context and choose variables to represent the quantities, which are governed by the constraint equation and the fact that they are non-negative (allowing us to restrict the graphs to points in the first quadrant only).

The four parts are independent and can be used as separate tasks.

Type: Problem-Solving Task

The purpose of this task is to help students learn to read information about a function from its graph, by asking them to show the part of the graph that exhibits a certain property of the function. The task could be used to further instruction on understanding functions or as an assessment tool, with the caveat that it requires some amount of creativity to decide how to best illustrate some of the statements.

Type: Problem-Solving Task

This task presents a simple but mathematically interesting game whose solution is a challenging exercise in creating and reasoning with algebraic inequalities. The core of the task involves converting a verbal statement into a mathematical inequality in a context in which the inequality is not obviously presented, and then repeatedly using the inequality to deduce information about the structure of the game.

Type: Problem-Solving Task

Students are given a word problem that can be solved by using a pair of linear equations. This task does not actually require that the student solve the system but that they recognize the pairs of linear equations in two variables that would be used to solve the system. This is an important step in the process of solving systems.

Type: Problem-Solving Task

This task examines the ways in which the plane can be covered by regular polygons in a very strict arrangement called a regular tessellation. These tessellations are studied here using algebra, which enters the picture via the formula for the measure of the interior angles of a regular polygon (which should therefore be introduced or reviewed before beginning the task). The goal of the task is to use algebra in order to understand which tessellations of the plane with regular polygons are possible.

Type: Problem-Solving Task

In this example, students use properties of rational exponents and other algebraic concepts to compare and verify the relative size of two real numbers that involve decimal exponents.

Type: Problem-Solving Task

The purpose of this task is to illustrate through an absurd example the fact that in real life quantities are reported to a certain level of accuracy, and it does not make sense to treat them as having greater accuracy.

Type: Problem-Solving Task

This task operates at two levels. In part it is a simple exploration of the relationship between speed, distance, and time. Part (c) requires understanding of the idea of average speed, and gives an opportunity to address the common confusion between average speed and the average of the speeds for the two segments of the trip.

At a higher level, the task addresses MAFS.912.N-Q.1.3, since realistically neither the car nor the bus is going to travel at exactly the same speed from beginning to end of each segment; there is time traveling through traffic in cities, and even on the autobahn the speed is not constant. Thus students must make judgments about the level of accuracy with which to report the result.

Type: Problem-Solving Task

This task examines, from a mathematical and statistical point of view, how scientists measure the age of organic materials by measuring the ratio of Carbon 14 to Carbon 12. The focus here is on the statistical nature of such dating.

Type: Problem-Solving Task

This task examines, from a mathematical and statistical point of view, how scientists measure the age of organic materials by measuring the ratio of Carbon 14 to Carbon 12. The focus here is on the statistical nature of such dating.

Type: Problem-Solving Task

The problem requires students to not only convert miles to kilometers and gallons to liters but they also have to deal with the added complication of finding the reciprocal at some point.

Type: Problem-Solving Task

This task asks students to calculate the cost of materials to make a penny, utilizing rates of grams of copper.

Type: Problem-Solving Task

There are many different ways to write exponential expressions that describe the same quantity, in this task the amount of a radioactive substance after *t* years. Depending on what aspect of the context we need to investigate, one expression of the quantity may be more useful than another. This task contrasts the usefulness of four equivalent expressions. Students first have to confirm that the given expressions for the radioactive substance are equivalent. Then they have to explain the significance of each expression in the context of the situation.

Type: Problem-Solving Task

Students are asked to use units to determine if the given statement is valid.

Type: Problem-Solving Task

This is a challenging task, suitable for extended work, and reaching into a deep understanding of units. Students are given a scenario and asked to determine the number of people required to complete the amount of work in the time described. The task requires students to exhibit MAFS.K12.MP.1.1, Make sense of problems and persevere in solving them. An algebraic solution is possible but complicated; a numerical solution is both simpler and more sophisticated, requiring skilled use of units and quantitative reasoning. Thus the task aligns with either MAFS.912.A-CED.1.1 or MAFS.912.N-Q.1.1, depending on the approach.

Type: Problem-Solving Task

Students manipulate a given equation to find specified information.

Type: Problem-Solving Task

Students solve problems tracking the balance of a checking account used only to pay rent. This simple conceptual task focuses on what it means for a number to be a solution to an equation, rather than on the process of solving equations.

Type: Problem-Solving Task

Students extrapolate the list price of a car given a total amount paid in states with different tax rates. The emphasis in this task is not on complex solution procedures. Rather, the progression of equations, from two that involve different values of the sales tax, to one that involves the sales tax as a parameter, is designed to foster the habit of looking for regularity in solution procedures, so that students don't approach every equation as a new problem but learn to notice familiar types.

Type: Problem-Solving Task

In this resource, students refer to given information which defines 5 variables in the context of real world government expenses. They are then asked to write equations based upon specific known values for some of the variables. The emphasis is on setting up, rather than solving, the equations.

Type: Problem-Solving Task

In this resource, a method of deriving the quadratic formula from a theoretical standpoint is demonstrated. This task is for instructional purposes only and builds on "Building an explicit quadratic function."

Type: Problem-Solving Task

This task compares the usefulness of different forms of a quadratic expression. Students have to choose which form most easily provides information about the maximum value, the zeros and the vertical intercept of a quadratic expression in the context of a real world situation. Rather than just manipulating one form into the other, students can make sense out of the structure of the expressions.

(From *Algebra: Form and Function*, McCallum et al., Wiley 2010)

Type: Problem-Solving Task

The purpose of this task is to help students see manipulation of expressions as an activity undertaken for a purpose.

Variation 1 of this task presents a related more complex expression already in the correct form to answer the question.

The expression arises in physics as the reciprocal of the combined resistance of two resistors in parallel. However, the context is not explicitly considered here.

Type: Problem-Solving Task

This task illustrates the process of rearranging the terms of an expression to reveal different aspects about the quantity it represents, precisely the language being used in standard MAFS.912.A-SSE.2.3. Students are provided with an expression giving the temperature of a container at a time *t*, and have to use simple inequalities (e.g., that 2^{t}>0 for all *t*) to reduce the complexity of an expression to a form where bounds on the temperature of a container of ice cream are made apparent.

Type: Problem-Solving Task

Students explore and manipulate expressions based on the following statement:

A function f defined for -a < x < a is even if f(-x)=f(x) and is odd if f(-x)=-f(x) when -a < x < a. In this task we assume f is defined on such an interval, which might be the full real line (i.e., a=8).

Type: Problem-Solving Task

Students compare graphs of different quadratic functions, then produce equations of their own to satisfy given conditions.

This exploration can be done in class near the beginning of a unit on graphing parabolas. Students need to be familiar with intercepts, and need to know what the vertex is. It is effective after students have graphed parabolas in vertex form (y=a(x–h)^{2}+k), but have not yet explored graphing other forms.

Type: Problem-Solving Task

Students are asked to interpret the effect on the value of an expression given a change in value of one of the variables.

Type: Problem-Solving Task

Students examine and answer questions related to a scenario similar to a "mixture" problem involving two different mixtures of fertilizer. In this example, students determine and then compare expressions that correspond to concentrations of various mixtures. Ultimately, students generalize the problem and verify conclusions using algebraic rather than numerical expressions.

Type: Problem-Solving Task

Students are asked to interpret expressions and equations within the context of the amounts of caramels and truffles in a box of candy.

Type: Problem-Solving Task

This problem asks students to consider algebraic expressions calculating the number of floor tiles in given patterns. The purpose of this task is to give students practice in reading, analyzing, and constructing algebraic expressions, attending to the relationship between the form of an expression and the context from which it arises. The context here is intentionally thin; the point is not to provide a practical application to kitchen floors, but to give a framework that imbues the expressions with an external meaning.

Type: Problem-Solving Task

This resource describes a simple scenario which can be represented by the use of variables. Students are asked to examine several variable expressions, interpret their meaning, and describe what quantities they each represent in the given context.

Type: Problem-Solving Task

This resource poses the question, "how many vehicles might be involved in a traffic jam 12 miles long?"

This task, while involving relatively simple arithmetic, promps students to practice modeling (MP4), work with units and conversion (N-Q.1), and develop a new unit (N-Q.2). Students will also consider the appropriate level of accuracy to use in their conclusions (N-Q.3).

Type: Problem-Solving Task

In this task students interpret the relative size of variable expressions involving two variables in the context of a real world situation. All given expressions can be interpreted as quantities that one might study when looking at two animal populations.

Type: Problem-Solving Task

The purpose of this task is to identify the structure in the two algebraic expressions by interpreting them in terms of a geometric context. Students will have likely seen this type of process before, so the principal source of challenge in this task is to encourage a multitude and variety of approaches, both in terms of the geometric argument and in terms of the algebraic manipulation.

Type: Problem-Solving Task

The task is a modeling problem which ties in to financial decisions faced routinely by businesses, namely the balance between maintaining inventory and raising short-term capital for investment or re-investment in developing the business.

Type: Problem-Solving Task

This task provides students the opportunity to make use of units to find the gas needed (MAFS.912.N-Q.1.1). It also requires them to make some sensible approximations (e.g., 2.92 gallons is not a good answer to part (a)) and to recognize that Felicia's situation requires her to round up. Various answers to (a) are possible, depending on how much students think is a safe amount for Felicia to have left in the tank when she arrives at the gas station. The key point is for them to explain their choices. This task provides an opportunity for students to practice MAFS.K12.MP.2.1: Reason abstractly and quantitatively, and MAFS.K12.MP.3.1: Construct viable arguments and critique the reasoning of others.

Type: Problem-Solving Task

This task is designed to make students think about the meaning of the quantities presented in the context and choose which ones are appropriate for the two different constraints presented. In particular, note that the purpose of the task is to have students generate the constraint equations for each part (though the problem statements avoid using this particular terminology), and not to have students solve said equations. If desired, instructors could also use this task to touch on such solutions by finding and interpreting solutions to the system of equations created in parts (a) and (b).

Type: Problem-Solving Task

The primary purpose of this task is to lead students to a numerical and graphical understanding of the behavior of a rational function near a vertical asymptote, in terms of the expression defining the function.

Type: Problem-Solving Task

The purpose of this task is to give students practice constructing functions that represent a quantity of interest in a context, and then interpreting features of the function in the light of the context. It can be used as either an assessment or a teaching task.

Type: Problem-Solving Task

This problem involves the meaning of numbers found on labels. When the level of accuracy is not given we need to make assumptions based on how the information is reported. An unexpected surprise awaits in this case, however, as no reasonable interpretation of the level of accuracy makes sense of the information reported on the bottles in parts (b) and (c). Either a miscalculation has been made or the numbers have been rounded in a very odd way.

Type: Problem-Solving Task

## Student Center Activity

This video will demonstrate how to multiply polynomials.

Type: Student Center Activity

## Tutorials

This video will demonstrate how to simplify square roots involving variables.

Type: Tutorial

Students will investigate symmetry by rotating polygons 180 degrees about their center.

Type: Tutorial

Students are shown, with an interactive tool, how to reflect a line segment. Students should have an understanding of slope and midpoint before viewing this video.

Type: Tutorial

This tutorial uses the midpoint of two lines to find the line of reflection.

Type: Tutorial

Students will see what happens when a figure is rotated about the origin -270 degrees. Having a foundation about right triangles is recommended before viewing this video.

Type: Tutorial

This video is an example of solving systems by elimination where the system has infinite solutions.

Type: Tutorial

This video shows how to solve a system of equations through simple elimination.

Type: Tutorial

This video explains how to identify systems of equations without a solution.

Type: Tutorial

In this tutorial, students are introduced to the concept that three non-collinear points are necessary to define a unique plane.

Type: Tutorial

Before learning any new concept it's important students learn and use common language and label concepts consistently. This tutorial introduces students to th point, line and plane.

Type: Tutorial

This video will demonstrate how to solve radical equations with additional practice problems.

Type: Tutorial

This tutorial is great practice for help in identifying parallel and perpendicular lines.

Type: Tutorial

This video demonstrates Bhaskara's proof of the Pythagorean Theorem.

Type: Tutorial

This video visually proves the Pythagorean Theorem using triangles and parallelograms.

Type: Tutorial

This video shows a proof of the Pythagorean Theorem using similar triangles.

Type: Tutorial

This video tutorial shows students: the standard form of a polynomial, how to identify polynomials, how to determine the degree of a polynomial, how to add and subtract polynomials, and how to represent the area of a shape as an addition or subtraction of polynomials.

Type: Tutorial

This video shows how to solve systems by elimination.

Type: Tutorial

This tutorial will help the students to understand the function notation such as f(x), which can be thought as another way of representing the y-value in a function, especially when graphing. The y-axis is even labeled as the f(x) axis, when graphing.

Type: Tutorial

This video is an introduction to the elimination method of solving a system of equations.

Type: Tutorial

This example demonstrates solving a system of equations algebraically and graphically.

Type: Tutorial

This video demonstrates a system of equations with no solution.

Type: Tutorial

This video shows how to solve a system of equations using the substitution method.

Type: Tutorial

This video demonstrates solving a system of equations word problem using elimination.

Type: Tutorial

In this tutorial, students will learn how to solve and graph a system of equations.

Type: Tutorial

This tutorial shows students how to solve and graph a system of equations. Students will see how to sketch their solution after solving the system of equations.

Type: Tutorial

This tutorial shows how to solve a system of equations by graphing. Students will see what a no solution system of equations looks like in a graph.

Type: Tutorial

This tutorial shows how to solve a system of equations using substitution.

Type: Tutorial

This video will demonstrate how to graph the solution to a system of inequalities.

Type: Tutorial

Many real world problems involve involve percentages. This lecture shows how algebra is used in solving problems of percent change and profit-and-loss.

Type: Tutorial

In this video, you will learn about Rene Descartes, and how he bridged the gap between algebra and geometry.

Type: Tutorial

<p>In this tutorial we will learn the basics of geometry, such as identifying a line, ray, point, and segment.</p>

Type: Tutorial

This video explains how to subtract polynomials with multiple variables and reinforces how to distribute a negative number.

Type: Tutorial

This video covers squaring a binomial with two variables. Students will be given the area of a square.

Type: Tutorial

It's helpful to represent an equation on a graph where we plot at least 2 points to show the relationship between the dependent and independent variables. Watch and we'll show you.

Type: Tutorial

This video tutorial discusses how to create a system of equations.

Type: Tutorial

This 8 minute video will show step-by-step directions for using the elimination method to solve a system of linear equations.

Type: Tutorial

This video provides a real-world scenario and step-by-step instructions to constructing equations using two variables. Possible follow-up videos include *Plotting System of Equations - Yoga Plan, Solving System of Equations with Substitution - Yoga Plan, and Solving System of Equations with Elimination - Yoga Plan.*

Type: Tutorial

This tutorial will help the learners to graph the equation of the quadratic function using the coordinates of the vertex of a parabola adn its x- intercepts.

Type: Tutorial

Evaluating Expressions with Two Variables

Type: Tutorial

This tutorial will help you to learn about the exponential functions by graphing various equations representing exponential growth and decay.

Type: Tutorial

In this example we have a formula for converting Celsius temperature to Fahrenheit. Let's substitute the variable with a value (Celsius temp) to get the degrees in Fahrenheit. Great problem to practice with us!

Type: Tutorial

Learn how to evaluate an expression with variables using a technique called substitution (or "plugging in").

Type: Tutorial

This video discusses multiplication and division of inequalities with negative numbers to solve the inequality.

Type: Tutorial

Great question. In algebra, we do indeed avoid using the multiplication sign. We'll explain it for you here.

Type: Tutorial

Our focus here is understanding that a variable is just a letter or symbol (usually a lower case letter) that can represent different values in an expression. We got this. Just watch.

Type: Tutorial

This tutorial demonstrates how to use the power of a power property with both numerals and variables.

Type: Tutorial

This lecture shows how algebra is used to solve problems involving mixtures of solutions of different concentrations.

Type: Tutorial

When solving a system of linear equations in x and y with a single solution, we get a unique pair of values for x and y. But what happens when try to solve a system with no solutions or an infinite number of solutions?

Type: Tutorial

Systems of two linear equations in two variables can have a single solution, no solutions, or an infinite number of solutions. This video gives a great description of inconsistent, dependent, and independent systems. A consistent independent system of equations will have one solution. A consistent dependent system of equations will have infinite number of solutions, and an inconsistent system of equations will have no solution. This tutorial also provides information on how to distinguish a given system of linear equations as inconsistent, independent, or dependent system by looking at the slope and intercept.

Type: Tutorial

A graph in Cartesian coordinates may represent a function or may only represent a binary relation. The "vertical line test" is a visual way to determine whether or not a graph represents a function.

Type: Tutorial

Systems of two equations in x and y can be solved by adding the equations to create a new equation with one variable eliminated. This new equation can then be solved to find the value of the remaining variable. That value is then substituted into either equation to find the value of other variable.

Type: Tutorial

A system of two equations in x and y can be solved by rearranging one equation to represent x in terms of y, and "substituting" this expression for x in the other equation. This creates an equation with only y which can then be solved to find y's value. This value can then be substituted into either equation to find the value of x.

Type: Tutorial

The video tutorial discusses about two typical polynomial multiplications. First, squaring a binomial and second, product of a sum and difference.

Type: Tutorial

This tutorial will help the learners practice multiplication of polynomials. Learners will understand that when they multiply expressions with more than two terms, they need to make sure each term in the first expression multiplies every term in the second expression.

Type: Tutorial

Binomials are the polynomials with two terms. This tutorial will help the students learn about the multiplication of binomials. In multiplication, we need to make sure that each term in the first set of parenthesis multiplies each term in the second set.

Type: Tutorial

In this tutorial, students will learn how to add and subtract polynomials functions using horizontal and vertical methods. In a horizontal format, like terms should be grouped together using the commutative property. In vertical format, terms should be listed by ascending degree with like terms placed below each other.

Type: Tutorial

In this tutorial students learn how to identify a polynomial, how to find the degree of a polynomial, and how to write a polynomial in standard format.

Type: Tutorial

This lesson introduces students to linear equations in one variable, shows how to solve them using addition, subtraction, multiplication, and division properties of equalities, and allows students to determine if a value is a solution, if there are infinitely many solutions, or no solution at all. The site contains an explanation of equations and linear equations, how to solve equations in general, and a strategy for solving linear equations. The lesson also explains contradiction (an equation with no solution) and identity (an equation with infinite solutions). There are five practice problems at the end for students to test their knowledge with links to answers and explanations of how those answers were found. Additional resources are also referenced.

Type: Tutorial

Upon completing this lesson, the student should be able to use the addition, subtraction, multiplication, and division properties of equality to solve linear inequalities, write the answer to an inequality using interval notation and draw a graph to give a visual answer to an inequality problem.

The lesson begins with explanations of inequality signs and interval notation and then moves on to demonstrate addition/subtraction and multiplication/division properties of equality. The site demonstrates a strategy for solving linear inequalities and presents three problems for students to practice what they have learned.

There is also a link to a previous tutorial which covers solving linear equations of one variable for students who need the review.

Type: Tutorial

## Video/Audio/Animations

This video will demonstrate how to determine what is and is not a function.

Type: Video/Audio/Animation

This video demonstrates how to determine if a relation is a function and how to identify the domain.

Type: Video/Audio/Animation

Although the domain and codomain of functions can consist of any type of objects, the most common functions encountered in Algebra are real-valued functions of a real variable, whose domain and codomain are the set of real numbers, R.

Type: Video/Audio/Animation

Exponents are not only integers and unit fractions. An exponent can be any rational number expressed as the quotient of two integers.

Type: Video/Audio/Animation

Radical expressions can often be simplified by moving factors which are perfect roots out from under the radical sign.

Type: Video/Audio/Animation

Mixture problems can involve mixtures of things other than liquids. This video shows how Algebra can be used to solve problems involving mixtures of different types of items.

Type: Video/Audio/Animation

When should a system of equations with multiple variables be used to solve an Algebra problem, instead of using a single equation with a single variable?

Type: Video/Audio/Animation

The points of intersection of two graphs represent common solutions to both equations. Finding these intersection points is an important tool in analyzing physical and mathematical systems.

Type: Video/Audio/Animation

This chapter presents a new look at the logic behind adding equations- the essential technique used when solving systems of equations by elimination.

Type: Video/Audio/Animation

Two sets which are often of primary interest when studying binary relations are the domain and range of the relation.

Type: Video/Audio/Animation

"Slope" is a fundamental concept in mathematics. Slope is often defined as " the rise over the run"....but why?

Type: Video/Audio/Animation

Th point-slope form of the equation for a line can describe any non-vertical line in the Cartesian plane, given the slope and the coordinates of a single point which lies on the line.

Type: Video/Audio/Animation

The two point form of the equation for a line can describe any non-vertical line in the Cartesian plane, given the coordinates of two points which lie on the line.

Type: Video/Audio/Animation

Literal equations are formulas for calculating the value of one unknown quantity from one or more known quantities. Variables in the formula are replaced by the actual or 'literal' values corresponding to a specific instance of the relationship.

Type: Video/Audio/Animation

This video illustrates how to determine if the graphs of a given set of equations are parallel.

Type: Video/Audio/Animation

This video takes a look at rearranging a formula to highlight a quantity of interest.

Type: Video/Audio/Animation

This video demonstrates writing a function that represents a real-life scenario.

Type: Video/Audio/Animation

Khan Academy video tutorial on graphing linear equations: "Algebra: Graphing Lines 1"

Type: Video/Audio/Animation

This resource gives an animated and then annotated proof of the Pythagorean Theorem.

Type: Video/Audio/Animation

This Khan Academy video tutorial introduces averages and algebra problems involving averages.

Type: Video/Audio/Animation

## Virtual Manipulatives

This interactive manipulative will let you compare and calculate volumes of different solids.

Type: Virtual Manipulative

This geogebratube interactive worksheet shows the step by step process for inscribing a regular hexagon in a circle. There are other geogebratube interactive worksheets for the square and the equilateral triangle.

Type: Virtual Manipulative

Using this resource, students can manipulate the measurements of a 3-D hourglass figure (double-napped cone) and its intersecting plane to see how the graph of a conic section changes. Students will see the impact of changing the *height* and *slant* of the cone and the *m* and *b* values of the plane on the shape of the graph. Students can also rotate and re-size the cone and graph to view from different angles.

Type: Virtual Manipulative

This resource will assess students' understanding of addition and subtraction of polynomials.

Type: Virtual Manipulative

This virtual manipulative is intended to allow the student to practice multiplication of binomials. The student should understand how to use algebra tiles before using this tool.

Type: Virtual Manipulative

This virtual manipulative allows the user to explore scenarios of a loan repayment by manipulating the amount of the loan, interest rate, payment amount, frequency of payments, and length of the loan in years.

Type: Virtual Manipulative

This virtual manipulative allows students to manipulate blocks, add or remove blocks, and connect them together to form solids. They can also experiment with counting the number of exposed faces, seeing what happens to the surface area when blocks are added or removed, and "unfolding" a block to create a net .

Type: Virtual Manipulative

This simulation allows the student to create a box plot and a histogram for the same set of data and toggle between the two displays. Maximum, minimum, median and mean are shown for the data set. The student can change the cell width to explore how the histogram is affected.

Type: Virtual Manipulative

In this activity, students adjust slider bars which adjust the coefficients and constants of a linear function and examine how their changes affect the graph. The equation of the line can be in slope-intercept form or standard form. This activity allows students to explore linear equations, slopes, and y-intercepts and their visual representation on a graph. This activity includes supplemental materials, including background information about the topics covered, a description of how to use the application, and exploration questions for use with the java applet.

Type: Virtual Manipulative

With this online Java applet, students use slider bars to move a cross section of a cone, cylinder, prism, or pyramid. This activity allows students to explore conic sections and the 3-dimensional shapes from which they are derived. This activity includes supplemental materials, including background information about the topics covered, a description of how to use the application, and exploration questions for use with the java applet.

Type: Virtual Manipulative

Allows students access to a Cartesian Coordinate System where linear equations can be graphed and details of the line and the slope can be observed.

Type: Virtual Manipulative

In this activity, students use preset data or enter in their own data to be represented in a box plot. This activity allows students to explore single as well as side-by-side box plots of different data. This activity includes supplemental materials, including background information about the topics covered, a description of how to use the application, and exploration questions for use with the Java applet.

Type: Virtual Manipulative

Using this virtual manipulative, students are able to graph a function and a set of ordered pairs on the same coordinate plane. The constants, coefficients, and exponents can be adjusted using slider bars, so the student can explore the affect on the graph as the function parameters are changed. Students can also examine the deviation of the data from the function. This activity includes supplemental materials, including background information about the topics covered, a description of how to use the application, and exploration questions for use with the java applet.

Type: Virtual Manipulative

With this online tool, students adjust the standard deviation and sample size of a normal distribution to see how it will affect a histogram of that distribution. This activity allows students to explore the effect of changing the sample size in an experiment and the effect of changing the standard deviation of a normal distribution. Tabs at the top of the page provide access to supplemental materials, including background information about the topics covered, a description of how to use the application, and exploration questions for use with the java applet.

Type: Virtual Manipulative

In this online tool, students input a function to create a graph where the constants, coefficients, and exponents can be adjusted by slider bars. This tool allows students to explore graphs of functions and how adjusting the numbers in the function affect the graph. Using tabs at the top of the page you can also access supplemental materials, including background information about the topics covered, a description of how to use the application, and exploration questions for use with the java applet.

Type: Virtual Manipulative

This is an online graphing utility that can be used to create box plots, bubble graphs, scatterplots, histograms, and stem-and-leaf plots.

Type: Virtual Manipulative

This web address, from the National Library of Virtual Manipulatives, will help teachers and students validate the Pythagorean Theorem both geometrically and algebraically. It can be used interactively with the Smartboard and the Promethean Board to create a better understanding of the topic.

Type: Virtual Manipulative

This interactive simulation investigates graphing linear and quadratic equations. Users are given the ability to define and change the coefficients and constants in order to observe resulting changes in the graph(s).

Type: Virtual Manipulative

This site provides a virtual balance on which the student can represent (and then solve) simple linear equations with integer answers. Conceptually, positive weights (unit-blocks and x-boxes) push the pans of the scale downward. Negative values are represented by balloons which can be attached to the pans of the scale. The student can then manipulate the weights to solve the equation while simultaneously seeing a visual display of these effects on the equation.

Type: Virtual Manipulative

The user clicks and drags a shape they have constructed to view its reflection across a line. A background grid and axes may or may not be used. The reflection may by examined analytically using coordinates. Symmetry may be displayed.

Type: Virtual Manipulative

Rotate shapes and their images with or without a background grid and axes.

Type: Virtual Manipulative

This program allows users to explore spatial geometry in a dynamic and interactive way. The tool allows users to rotate, zoom out, zoom in, and translate a plethora of polyhedra. The program is able to compute topological and geometrical duals of each polyhedron. Geometrical operations include unfolding, plane sections, truncation, and stellation.

Type: Virtual Manipulative

This virtual manipulative histogram tool can aid in analyzing the distribution of a dataset. It has 6 preset datasets and a function to add your own data for analysis.

Type: Virtual Manipulative

This activity allows the user to graph data sets in multiple bar graphs. The color, thickness, and scale of the graph are adjustable which may produce graphs that are misleading. Users may input their own data, or use or alter pre-made data sets. This activity includes supplemental materials, including background information about the topics covered, a description of how to use the application, and exploration questions for use with the java applet.

Type: Virtual Manipulative

In this activity, students can create and view a histogram using existing data sets or original data entered. Students can adjust the interval size using a slider bar, and they can also adjust the other scales on the graph. This activity allows students to explore histograms as a way to represent data as well as the concepts of mean, standard deviation, and scale. This activity includes supplemental materials, including background information about the topics covered, a description of how to use the application, and exploration questions for use with the java applet.

Type: Virtual Manipulative

## Parent Resources

## Perspectives Video: Expert

It's important to stay inside the lines of your project constraints to finish in time and under budget. This NASA systems engineer explains how constraints can actually promote creativity and help him solve problems!

Download the CPALMS Perspectives video student note taking guide.

Type: Perspectives Video: Expert

## Perspectives Video: Professional/Enthusiasts

Listen in as a computing enthusiast describes how hexadecimal notation is used to express big numbers in just a little space.

Download the CPALMS Perspectives video student note taking guide.

Type: Perspectives Video: Professional/Enthusiast

Get fired up as you learn more about ceramic glaze recipes and mathematical units.

Download the CPALMS Perspectives video student note taking guide.

Type: Perspectives Video: Professional/Enthusiast

Don't be a shrinking violet. Learn how uniform scaling is important for candy production.

Download the CPALMS Perspectives video student note taking guide.

Type: Perspectives Video: Professional/Enthusiast

See and see far into the future of arts and manufacturing as a technician explains computer numerically controlled (CNC) machining bit by bit.

Download the CPALMS Perspectives video student note taking guide.

Type: Perspectives Video: Professional/Enthusiast

You'll need to bring your computer skills and math knowledge to estimate oil volume and rate as it seeps from the ocean floor. Dive in!

Download the CPALMS Perspectives video student note taking guide.

Type: Perspectives Video: Professional/Enthusiast

## Problem-Solving Tasks

Students explore the structure of the operation *s*/(v*n*). This question provides students with an opportunity to see expressions as constructed out of a sequence of operations: first taking the square root of *n*, then dividing the result of that operation into *s*.

Type: Problem-Solving Task

The purpose of this task is to allow students to demonstrate an ability to construct boxplots and to use boxplots as the basis for comparing distributions.

Type: Problem-Solving Task

This problem solving task challenges students to answer probability questions about SAT scores, using distribution and mean to solve the problem.

Type: Problem-Solving Task

This problem could be used as an introductory lesson to introduce group comparisons and to engage students in a question they may find amusing and interesting.

Type: Problem-Solving Task

The purpose of this task is to have students complete normal distribution calculations and to use properties of normal distributions to draw conclusions.

Type: Problem-Solving Task

This task requires students to use the normal distribution as a model for a data distribution. Students must use given means and standard deviations to approximate population percentages.

Type: Problem-Solving Task

The task provides a context to calculate discrete probabilities and represent them on a bar graph.

Type: Problem-Solving Task

This task asks students to use similarity to solve a problem in a context that will be familiar to many, though most students are accustomed to using intuition rather than geometric reasoning to set up the shot.

Type: Problem-Solving Task

In this problem, students are given a picture of two triangles that appear to be similar, but whose similarity cannot be proven without further information. Asking students to provide a sequence of similarity transformations that maps one triangle to the other, using the definition of similarity in terms of similarity transformations.

Type: Problem-Solving Task

This task involves a reasonably direct application of similar triangles, coupled with a moderately challenging procedure of constructing a diagram from a verbal description.

Type: Problem-Solving Task

The purpose of this task is to engage students in geometric modeling, and in particular to deduce algebraic relationships between variables stemming from geometric constraints.

Type: Problem-Solving Task

In this example, students are asked to write a function describing the population growth of algae. It is implied that this is exponential growth.

Type: Problem-Solving Task

Using a chart of diameters of different denominations of coins, students are asked to figure out how many coins fit around a central coin.

Type: Problem-Solving Task

This problem asks students to model phenomena on the surface of the earth by examining the visibility of the lamp in a lighthouse from a boat.

Type: Problem-Solving Task

This problem solving task encourages students to explore why solar eclipses are rare by examining the radius of the sun and the furthest distance between the moon and the earth.

Type: Problem-Solving Task

Using a triangle with line through it, students are tasked to show the congruent angles, and conclude if one triangle is similar to the other.

Type: Problem-Solving Task

This provides an opportunity to model a concrete situation with mathematics. Once a representative picture of the situation described in the problem is drawn (the teacher may provide guidance here as necessary), the solution of the task requires an understanding of the definition of the sine function.

Type: Problem-Solving Task

The goal of this task is to model a familiar object, an Olympic track, using geometric shapes. Calculations of perimeters of these shapes explain the staggered start of runners in a 400 meter race.

Type: Problem-Solving Task

In this problem, geometry is applied to a 400 meter track to find the perimeter of the track.

Type: Problem-Solving Task

In this task, a typographic grid system serves as the background for a standard paper clip. A metric measurement scale is drawn across the bottom of the grid and the paper clip extends in both directions slightly beyond the grid. Students are given the approximate length of the paper clip and determine the number of like paper clips made from a given length of wire.

Type: Problem-Solving Task

In this task, students will provide a sketch of a paper ice cream cone wrapper, use the sketch to develop a formula for the surface area of the wrapper, and estimate the maximum number of wrappers that could be cut from a rectangular piece of paper.

Type: Problem-Solving Task

This problem solving task asks students to explain which measurements are needed to estimate the thickness of a soda can.

Type: Problem-Solving Task

This problem solving task challenges students to find the surface area of a soda can, calculate how many cubic centimeters of aluminum it contains, and estimate how thick it is.

Type: Problem-Solving Task

This is a mathematical modeling task aimed at making a reasonable estimate for something which is too large to count accurately, the number of leaves on a tree.

Type: Problem-Solving Task

Type: Problem-Solving Task

This problem solving task challenges students to apply the concepts of mass, volume, and density in the real-world context to find how many cells are in the human body.

Type: Problem-Solving Task

The goal of this task is to use geometry to study the structure of beehives.

Type: Problem-Solving Task

Reflective of the modernness of the technology involved, this is a challenging geometric modeling task in which students discover from scratch the geometric principles underlying the software used by GPS systems.

Type: Problem-Solving Task

This problem solving task gives an interesting context for implementing ideas from geometry and trigonometry.

Type: Problem-Solving Task

This problem solving task uses the tale of Archimedes and the King of Syracuse's crown to determine the volume and mass of gold and silver.

Type: Problem-Solving Task

This problem solving task asks students to find the area of a triangle by using unit squares and line segments.

Type: Problem-Solving Task

The purpose of the task is to analyze a plausible real-life scenario using a geometric model. The task requires knowledge of volume formulas for cylinders and cones, some geometric reasoning involving similar triangles, and pays attention to reasonable approximations and maintaining reasonable levels of accuracy throughout.

Type: Problem-Solving Task

This task provides a concrete geometric setting in which to study rigid transformations of the plane.

Type: Problem-Solving Task

This task provides an opportunity for students to apply triangle congruence theorems in an explicit, interesting context.

Type: Problem-Solving Task

This problem solving task challenges students to inscribe equilateral triangles and regular hexagons on a circle with a compass and straightedge.

Type: Problem-Solving Task

This problem solving task challenges students to construct a perpendicular bisector of a given segment.

Type: Problem-Solving Task

This task asks students to use a straightedge and compass to construct the line across which a triangle is reflected.

Type: Problem-Solving Task

The purpose of this task is to use geometric and algebraic reasoning to model a real-life scenario. In particular, students are in several places (implicitly or explicitly) to reason as to when making approximations is reasonable and when to round, when to use equalities vs. inequalities, and the choice of units to work with (e.g., mm vs. cm).

Type: Problem-Solving Task

This task presents a context that leads students toward discovery of the formula for calculating the volume of a sphere.

Type: Problem-Solving Task

This task is inspired by the derivation of the volume formula for the sphere. If a sphere of radius 1 is enclosed in a cylinder of radius 1 and height 2, then the volume not occupied by the sphere is equal to the volume of a "double-naped cone" with vertex at the center of the sphere and bases equal to the bases of the cylinder

Type: Problem-Solving Task

This problem solving task challenges students to bisect a given angle.

Type: Problem-Solving Task

This problem solving task challenges students to place a warehouse (point) an equal distance from three roads (lines).

Type: Problem-Solving Task

This problem solving task challenges students to find the perpendicular meeting point of a segment from the center of a circle and a tangent.

Type: Problem-Solving Task

This task provides a construction of the angle bisector of an angle by reducing it to the bisection of an angle to finding the midpoint of a line segment. It is worth observing the symmetry -- for both finding midpoints and bisecting angles, the goal is to cut an object into two equal parts.

Type: Problem-Solving Task

In this task, students use trigonometric functions to model the movement of a point around a wheel and, through space. Students also interpret features of graphs in terms of the given real-world context.

Type: Problem-Solving Task

This problem solving task challenges students to find the linear, exponential and quadratic functions based on two points.

Type: Problem-Solving Task

This problem solving task asks students to predict and model US population based on a chart of US population data from 1982 to 1988.

Type: Problem-Solving Task

This problem solving task asks students to solve five exponential and linear function problems based on a US population chart for the years 1790-1860.

Type: Problem-Solving Task

This simple conceptual problem does not require algebraic manipulation, but requires students to articulate the reasoning behind each statement.

Type: Problem-Solving Task

The purpose of this task is to give students an opportunity to explore various aspects of exponential models (e.g., distinguishing between constant absolute growth and constant relative growth, solving equations using logarithms, applying compound interest formulas) in the context of a real world problem with ties to developing financial literacy skills. In particular, students are introduced to the idea of inflation of prices of a single commodity, and are given a very brief introduction to the notion of the Consumer Price Index for measuring inflation of a body of goods.

Type: Problem-Solving Task

This task gives a variation of real-life contexts which could be modeled by a linear or exponential function. The key distinguishing feature between the two is whether the change by equal factors over equal intervals (exponential functions), or by a constant increase per unit interval (linear functions). The task could either be used as an assessment problem on this distinction, or used as an introduction to the differences between these very important classes of functions.

Type: Problem-Solving Task

This task requires students to use the fact that on the graph of the linear function h(x) = ax + b, the y-coordinate increases by a when x increases by one. Specific values for a and b were left out intentionally to encourage students to use the above fact as opposed to computing the point of intersection, (p,q), and then computing respective function values to answer the question.

Type: Problem-Solving Task

The typical system of equations or inequalities problem gives the system and asks for the graph of the solution. This task turns the problem around. It gives a solution set and asks for the system that corresponds to it. The purpose of this task is to give students a chance to go beyond the typical problem and make the connections between points in the coordinate plane and solutions to inequalities and equations. Students have to focus on what the graph is showing. When you are describing a region, why does the inequality have to go one way or another? When you pick a point that clearly lies in a region, what has to be true about its coordinates so that it satisfies the associated system of inequalities?

Type: Problem-Solving Task

In order to engage this task meaningfully, students must be aware of the convention that va for a positive number a refers to the positive square root of a. The purpose of the task is to show students a situation where squaring both sides of an equation can result in an equation with more solutions than the original one.

Type: Problem-Solving Task

This task is an example of a mathematical modeling problem (SMP 4) and it also illustrates SMP 1 (Making sense of a problem). Students are only told that there are two ingredients in the pasta and they have a picture of the box. It might even be better to just show the picture of the box, or to bring in the box and ask the students to pose the question themselves. The brand of pasta is quite commonly available at supermarkets or health food stores such as Whole Foods and even at Amazon.com. The box has the nutritional label and a reference to the website where the students can find other information about the ingredients

Type: Problem-Solving Task

This task has some aspects of a mathematical modeling problem (SMP 4) and it also illustrates SMP 1 (Making sense of a problem). Students are given all the relevant information on the nutritional labels, but they have to figure out how to use this information. They have to come up with the idea that they can set up two equations in two unknowns to solve the problem.

Type: Problem-Solving Task

This task addresses A-REI.3.6, solving systems of linear equations exactly, and provides a simple example of a system with three equations and three unknown. Two (of many) methods for solving the system are presented. The first takes the given information to make three equations in three unknowns which can then be solved via algebraic manipulation to find the three numbers. The second solution is more clever, creating a single equation in three unknowns from the given information. This equation is then combined with the given information about the sums of pairs of numbers to deduce what the third number is. In reality, this solution is not simpler than the first: rather it sets up a slightly different set of equations which can be readily solved (the key being to take the sum of the three equations in the first solution). It provides a good opportunity for the instructor to show different methods for solving the same system of linear equations.

Type: Problem-Solving Task

The purpose of this task is to continue a crucial strand of algebraic reasoning begun at the middle school level (e.g, 6.EE.5). By asking students to reason about solutions without explicitly solving them, we get at the heart of understanding what an equation is and what it means for a number to be a solution to an equation. The equations are intentionally very simple; the point of the task is not to test technique in solving equations, but to encourage students to reason about them.

Type: Problem-Solving Task

This task gives students an opportunity to work with exponential functions in a real world context involving continuously compounded interest. They will study how the base of the exponential function impacts its growth rate and use logarithms to solve exponential equations.

Type: Problem-Solving Task

This exploratory task requires the student to use properties of exponential functions in order to estimate how much Carbon 14 remains in a preserved plant after different amounts of time.

Type: Problem-Solving Task

This task involves a fairly straightforward decaying exponential. Filling out the table and developing the general formula is complicated only by the need to work with a fraction that requires decisions about rounding and precision.

Type: Problem-Solving Task

In this task students use verbal descriptions to construct and compare linear and exponential functions and to find where the two functions intersect (F-LE.2, F-LE.3, A-REI.11).

Type: Problem-Solving Task

This task is the last in a series of three tasks that use inequalities in the same context at increasing complexity in 6th grade, 7th grade and in HS algebra. Students write and solve inequalities, and represent the solutions graphically.

Type: Problem-Solving Task

The given solutions for this task involve the creation and solving of a system of two equations and two unknowns, with the caveat that the context of the problem implies that we are interested only in non-negative integer solutions. Indeed, in the first solution, we must also restrict our attention to the case that one of the variables is further even. This aspect of the task is illustrative of mathematical practice standard MP4 (Model with mathematics), and crucial as the system has an integer solution for both situations, that is, whether or not we include the dollar on the floor in the cash box or not.

Type: Problem-Solving Task

This task is a somewhat more complicated version of "Accurately weighing pennies I'' as a third equation is needed in order to solve part (a) explicitly. Instead, students have to combine the algebraic techniques with some additional problem-solving (numerical reasoning, informed guess-and-check, etc.) Part (b) is new to this task, as with only two types of pennies the weight of the collection determines how many pennies of each type are in the collection. This is no longer the case with three different weights but in this particular case, a collection of 50 is too small to show any ambiguity. This is part of the reason for part (c) of the question where the weight alone no longer determines which type of pennies are in the roll. This shows how important levels of accuracy in measurement are as the answer to part (b) could be different if we were to measure on a scale which is only accurate to the nearest tenth of a gram instead of to the nearest hundredth of a gram.

Type: Problem-Solving Task

This classroom task is meant to elicit a variety of different methods of solving a quadratic equation (A-REI.4). Some are straightforward (for example, expanding the square on the right and rearranging the equation so that we can use the quadratic formula); some are simple but clever (reasoning from the fact that x and (2x - 9) have the same square); some use tools (using a graphing calculator to graph the functions f(x) = x^2 and g(x) = (2x-90)^2 and looking for values of x at which the two functions intersect). Some solution methods will work on an arbitrary quadratic equation, while others (such as the last three) may have difficulty or fail if the quadratic equation is not given in a particular form, or if the solutions are not rational numbers.

Type: Problem-Solving Task

This problem involves solving a system of algebraic equations from a context: depending how the problem is interpreted, there may be one equation or two. The main work in parts (a) and (b) is in setting up the equation(s) appropriately. Question (c) is more subtle and it requires thinking carefully about the accuracy available in a particular measurement (weight). The first two parts of this task could be used for instructional or assessment purposes while the third part should strictly be implemented for instructional purposes.

Type: Problem-Solving Task

The purpose of this task is to provide an opportunity for students to reason about equivalence of equations. The instruction to give reasons that do not depend on solving the equation is intended to focus attention on the transformation of equations as a deductive step.

Type: Problem-Solving Task

This task requires students to use the fact that the value of an exponential function f(x) = a · b^x increases by a multiplicative factor of b when x increases by one. It intentionally omits specific values for c and d in order to encourage students to use this fact instead of computing the point of intersection, (p,q), and then computing function values to answer the question.

Type: Problem-Solving Task

This problem assumes that students are familiar with the notation x_{0} and Δx. However, the language "successive quotient" may be new.

Type: Problem-Solving Task

This task assumes that students are familiar with the ?x and ?y notations. Students most likely developed this familiarity in their work with slope.

Type: Problem-Solving Task

An important property of linear functions is that they grow by equal differences over equal intervals. In F.LE Equal Differences over Equal Intervals 1, students prove this for equal intervals of length one unit, and note that in this case the equal differences have the same value as the slope.

Type: Problem-Solving Task

This problem-solving task asks students to examine if linear modeling would be appropriate to describe and predict population growth from select years.

Type: Problem-Solving Task

This problem-solving task provides students an opportunity to experiment with modeling real data by using population growth rates from the past two centuries.

Type: Problem-Solving Task

This problem-solving task challenges students to write expressions and create a table to calculate how much money can be gained after investing at different banks with different interest rates.

Type: Problem-Solving Task

This problem-solving task asks students to describe exponential growth through a real-world problem involving the illegal introduction of fish into a lake.

Type: Problem-Solving Task

This problem-solving emphasizes the expectation that students know linear functions grow by constant differences over equal intervals and exponential functions grow by constant factors over equal intervals.

Type: Problem-Solving Task

This is a simple task touching on two key points of functions. First, there is the idea that not all functions have real numbers as domain and range values. Second, the task addresses the issue of when a function admits an inverse, and the process of "restricting the domain" in order to achieve an invertible function.

Type: Problem-Solving Task

The purpose of this task is to give students practice interpreting statements using function notation. It can be used as a diagnostic if students seem to be having trouble with function notation, for example mistakenly interpreting f(x) as the product of f and x.

Type: Problem-Solving Task

This task is meant to be a straight-forward assessment task of graph reading and interpreting skills. This task helps reinforce the idea that when a variable represents time, t = 0 is chosen as an arbitrary point in time and positive times are interpreted as times that happen after that.

Type: Problem-Solving Task

This task addresses a common misconception about function notation.

Type: Problem-Solving Task

This task could be used for assessment or for practice. It allows students to compare characteristics of two quadratic functions that are each represented differently, one as the graph of a quadratic function and one written out algebraically. Specifically, students are asked to determine which function has the greatest maximum and the greatest non-negative root.

Type: Problem-Solving Task

This task requires interpreting a function in a non-standard context. While the domain and range of this function are both numbers, the way in which the function is determined is not via a formula but by a (pre-determined) sequence of coin flips. In addition, the task provides an opportunity to compute some probabilities in a discrete situation. The task could be used to segue the discussion from functions to probability, in particular the early standards in the S-CP domain.

Type: Problem-Solving Task

The purpose of this task is to investigate the meaning of the definition of function in a real-world context where the question of whether there is more than one output for a given input arises naturally. In more advanced courses this task could be used to investigate the question of whether a function has an inverse.

Type: Problem-Solving Task

The purpose of this task to help students think about an expression for a function as built up out of simple operations on the variable and understand the domain in terms of values for which each operation is invalid (e.g., dividing by zero or taking the square root of a negative number).

Type: Problem-Solving Task

This simple task assesses whether students can interpret function notation. The four parts of the task provide a logical progression of exercises for advancing understanding of function notation and how to interpret it in terms of a given context.

Type: Problem-Solving Task

This task asks students to find the average, write an equation, find the domain, and create a graph of the cost of producing DVDs.

Type: Problem-Solving Task

This problem provides students with an opportunity to discover algebraic structure in a geometric context. More specifically, the student will need to divide up the given polygons into triangles and then use the fact that the sum of the angles in each triangle is 180^{°}.

Type: Problem-Solving Task

The principal purpose of the task is to explore a real-world application problem with algebra, working with units and maintaining reasonable levels of accuracy throughout. Students are asked to determine which product will be the most economical to meet the requirements given in the problem.

Type: Problem-Solving Task

This task asks students to consider functions in regard to temperatures in a high school gym.

Type: Problem-Solving Task

The purpose of this task is to introduce or reinforce the concept of a function, especially in a context where the function is not given by an explicit algebraic representation. Further, the last part of the task emphasizes the significance of one variable being a function of another variable in an immediately relevant real-life context.

Type: Problem-Solving Task

In this task students are given graphs of quantities related to weather. The purpose of the task is to show that graphs are more than a collection of coordinate points; they can tell a story about the variables that are involved, and together they can paint a very complete picture of a situation, in this case the weather. Features in one graph, like maximum and minimum points, correspond to features in another graph. For example, on a rainy day, the solar radiation is very low, and the cumulative rainfall graph is increasing with a large slope.

Type: Problem-Solving Task

These problems form a bridge between work on functions and work on probability. The task is better suited for instruction than for assessment as it provides students with a non-standard setting in which to interpret the meaning of functions. Students should carry out the process of flipping a coin and modeling this Random Walk in order to develop a sense of the process before analyzing it mathematically.

Type: Problem-Solving Task

This task is designed to get at a common student confusion between the independent and dependent variables. This confusion often arises in situations like (b), where students are asked to solve an equation involving a function, and confuse that operation with evaluating the function.

Type: Problem-Solving Task

This tasks asks students to use functions to predict the price of a pizza on a specific day and find which day the pizza would be cheapest according to a promotion.

Type: Problem-Solving Task

This problem is a simple de-contextualized version of F-IF Your Father and F-IF Parking Lot. It also provides a natural context where the absolute value function arises as, in part (b), solving for x in terms of y means taking the square root of x^2 which is |x|.This task assumes students have an understanding of the relationship between functions and equations.

Type: Problem-Solving Task

This deceptively simple task asks students to find the domain and range of a function from a given context. The function is linear and if simply looked at from a formulaic point of view, students might find the formula for the line and say that the domain and range are all real numbers.

Type: Problem-Solving Task

This problem introduces a logistic growth model in the concrete settings of estimating the population of the U.S. The model gives a surprisingly accurate estimate and this should be contrasted with linear and exponential models.

Type: Problem-Solving Task

This task is for instructional purposes only and students should already be familiar with some specific examples of logistic growth functions. The goal of this task is to have students appreciate how different constants influence the shape of a graph.

Type: Problem-Solving Task

This task can be used as a quick assessment to see if students can make sense of a graph in the context of a real world situation. Students also have to pay attention to the scale on the vertical axis to find the correct match. The first and third graphs look very similar at first glance, but the function values are very different since the scales on the vertical axes are very different. The task could also be used to generate a group discussion on interpreting functions given by graphs.

Type: Problem-Solving Task

In this task, students will use inverse operations to solve the equations for the unknown variable or for the designated variable if there is more than one.

Type: Problem-Solving Task

The purpose of this task is to give students practice writing a constraint equation for a given context. Instruction accompanying this task should introduce the notion of a constraint equation as an equation governing the possible values of the variables in question (i.e., "constraining" said values). In particular, it is worth differentiating the role of constraint equations from more functional equations, e.g., formulas to convert from degrees Celsius to degree Fahrenheit. The task has students interpret the context and choose variables to represent the quantities, which are governed by the constraint equation and the fact that they are non-negative (allowing us to restrict the graphs to points in the first quadrant only).

The four parts are independent and can be used as separate tasks.

Type: Problem-Solving Task

The purpose of this task is to help students learn to read information about a function from its graph, by asking them to show the part of the graph that exhibits a certain property of the function. The task could be used to further instruction on understanding functions or as an assessment tool, with the caveat that it requires some amount of creativity to decide how to best illustrate some of the statements.

Type: Problem-Solving Task

This task presents a simple but mathematically interesting game whose solution is a challenging exercise in creating and reasoning with algebraic inequalities. The core of the task involves converting a verbal statement into a mathematical inequality in a context in which the inequality is not obviously presented, and then repeatedly using the inequality to deduce information about the structure of the game.

Type: Problem-Solving Task

Students are given a word problem that can be solved by using a pair of linear equations. This task does not actually require that the student solve the system but that they recognize the pairs of linear equations in two variables that would be used to solve the system. This is an important step in the process of solving systems.

Type: Problem-Solving Task

This task examines the ways in which the plane can be covered by regular polygons in a very strict arrangement called a regular tessellation. These tessellations are studied here using algebra, which enters the picture via the formula for the measure of the interior angles of a regular polygon (which should therefore be introduced or reviewed before beginning the task). The goal of the task is to use algebra in order to understand which tessellations of the plane with regular polygons are possible.

Type: Problem-Solving Task

In this example, students use properties of rational exponents and other algebraic concepts to compare and verify the relative size of two real numbers that involve decimal exponents.

Type: Problem-Solving Task

The purpose of this task is to illustrate through an absurd example the fact that in real life quantities are reported to a certain level of accuracy, and it does not make sense to treat them as having greater accuracy.

Type: Problem-Solving Task

This task operates at two levels. In part it is a simple exploration of the relationship between speed, distance, and time. Part (c) requires understanding of the idea of average speed, and gives an opportunity to address the common confusion between average speed and the average of the speeds for the two segments of the trip.

At a higher level, the task addresses MAFS.912.N-Q.1.3, since realistically neither the car nor the bus is going to travel at exactly the same speed from beginning to end of each segment; there is time traveling through traffic in cities, and even on the autobahn the speed is not constant. Thus students must make judgments about the level of accuracy with which to report the result.

Type: Problem-Solving Task

This task examines, from a mathematical and statistical point of view, how scientists measure the age of organic materials by measuring the ratio of Carbon 14 to Carbon 12. The focus here is on the statistical nature of such dating.

Type: Problem-Solving Task

Type: Problem-Solving Task

The problem requires students to not only convert miles to kilometers and gallons to liters but they also have to deal with the added complication of finding the reciprocal at some point.

Type: Problem-Solving Task

This task asks students to calculate the cost of materials to make a penny, utilizing rates of grams of copper.

Type: Problem-Solving Task

There are many different ways to write exponential expressions that describe the same quantity, in this task the amount of a radioactive substance after *t* years. Depending on what aspect of the context we need to investigate, one expression of the quantity may be more useful than another. This task contrasts the usefulness of four equivalent expressions. Students first have to confirm that the given expressions for the radioactive substance are equivalent. Then they have to explain the significance of each expression in the context of the situation.

Type: Problem-Solving Task

Students are asked to use units to determine if the given statement is valid.

Type: Problem-Solving Task

This is a challenging task, suitable for extended work, and reaching into a deep understanding of units. Students are given a scenario and asked to determine the number of people required to complete the amount of work in the time described. The task requires students to exhibit MAFS.K12.MP.1.1, Make sense of problems and persevere in solving them. An algebraic solution is possible but complicated; a numerical solution is both simpler and more sophisticated, requiring skilled use of units and quantitative reasoning. Thus the task aligns with either MAFS.912.A-CED.1.1 or MAFS.912.N-Q.1.1, depending on the approach.

Type: Problem-Solving Task

Students manipulate a given equation to find specified information.

Type: Problem-Solving Task

Students solve problems tracking the balance of a checking account used only to pay rent. This simple conceptual task focuses on what it means for a number to be a solution to an equation, rather than on the process of solving equations.

Type: Problem-Solving Task

Students extrapolate the list price of a car given a total amount paid in states with different tax rates. The emphasis in this task is not on complex solution procedures. Rather, the progression of equations, from two that involve different values of the sales tax, to one that involves the sales tax as a parameter, is designed to foster the habit of looking for regularity in solution procedures, so that students don't approach every equation as a new problem but learn to notice familiar types.

Type: Problem-Solving Task

In this resource, students refer to given information which defines 5 variables in the context of real world government expenses. They are then asked to write equations based upon specific known values for some of the variables. The emphasis is on setting up, rather than solving, the equations.

Type: Problem-Solving Task

In this resource, a method of deriving the quadratic formula from a theoretical standpoint is demonstrated. This task is for instructional purposes only and builds on "Building an explicit quadratic function."

Type: Problem-Solving Task

This task compares the usefulness of different forms of a quadratic expression. Students have to choose which form most easily provides information about the maximum value, the zeros and the vertical intercept of a quadratic expression in the context of a real world situation. Rather than just manipulating one form into the other, students can make sense out of the structure of the expressions.

(From *Algebra: Form and Function*, McCallum et al., Wiley 2010)

Type: Problem-Solving Task

The purpose of this task is to help students see manipulation of expressions as an activity undertaken for a purpose.

Variation 1 of this task presents a related more complex expression already in the correct form to answer the question.

The expression arises in physics as the reciprocal of the combined resistance of two resistors in parallel. However, the context is not explicitly considered here.

Type: Problem-Solving Task

This task illustrates the process of rearranging the terms of an expression to reveal different aspects about the quantity it represents, precisely the language being used in standard MAFS.912.A-SSE.2.3. Students are provided with an expression giving the temperature of a container at a time *t*, and have to use simple inequalities (e.g., that 2^{t}>0 for all *t*) to reduce the complexity of an expression to a form where bounds on the temperature of a container of ice cream are made apparent.

Type: Problem-Solving Task

Students explore and manipulate expressions based on the following statement:

A function f defined for -a < x < a is even if f(-x)=f(x) and is odd if f(-x)=-f(x) when -a < x < a. In this task we assume f is defined on such an interval, which might be the full real line (i.e., a=8).

Type: Problem-Solving Task

Students compare graphs of different quadratic functions, then produce equations of their own to satisfy given conditions.

This exploration can be done in class near the beginning of a unit on graphing parabolas. Students need to be familiar with intercepts, and need to know what the vertex is. It is effective after students have graphed parabolas in vertex form (y=a(x–h)^{2}+k), but have not yet explored graphing other forms.

Type: Problem-Solving Task

Students are asked to interpret the effect on the value of an expression given a change in value of one of the variables.

Type: Problem-Solving Task

Students examine and answer questions related to a scenario similar to a "mixture" problem involving two different mixtures of fertilizer. In this example, students determine and then compare expressions that correspond to concentrations of various mixtures. Ultimately, students generalize the problem and verify conclusions using algebraic rather than numerical expressions.

Type: Problem-Solving Task

Students are asked to interpret expressions and equations within the context of the amounts of caramels and truffles in a box of candy.

Type: Problem-Solving Task

This problem asks students to consider algebraic expressions calculating the number of floor tiles in given patterns. The purpose of this task is to give students practice in reading, analyzing, and constructing algebraic expressions, attending to the relationship between the form of an expression and the context from which it arises. The context here is intentionally thin; the point is not to provide a practical application to kitchen floors, but to give a framework that imbues the expressions with an external meaning.

Type: Problem-Solving Task

This resource describes a simple scenario which can be represented by the use of variables. Students are asked to examine several variable expressions, interpret their meaning, and describe what quantities they each represent in the given context.

Type: Problem-Solving Task

This resource poses the question, "how many vehicles might be involved in a traffic jam 12 miles long?"

This task, while involving relatively simple arithmetic, promps students to practice modeling (MP4), work with units and conversion (N-Q.1), and develop a new unit (N-Q.2). Students will also consider the appropriate level of accuracy to use in their conclusions (N-Q.3).

Type: Problem-Solving Task

In this task students interpret the relative size of variable expressions involving two variables in the context of a real world situation. All given expressions can be interpreted as quantities that one might study when looking at two animal populations.

Type: Problem-Solving Task

The purpose of this task is to identify the structure in the two algebraic expressions by interpreting them in terms of a geometric context. Students will have likely seen this type of process before, so the principal source of challenge in this task is to encourage a multitude and variety of approaches, both in terms of the geometric argument and in terms of the algebraic manipulation.

Type: Problem-Solving Task

The task is a modeling problem which ties in to financial decisions faced routinely by businesses, namely the balance between maintaining inventory and raising short-term capital for investment or re-investment in developing the business.

Type: Problem-Solving Task

This task provides students the opportunity to make use of units to find the gas needed (MAFS.912.N-Q.1.1). It also requires them to make some sensible approximations (e.g., 2.92 gallons is not a good answer to part (a)) and to recognize that Felicia's situation requires her to round up. Various answers to (a) are possible, depending on how much students think is a safe amount for Felicia to have left in the tank when she arrives at the gas station. The key point is for them to explain their choices. This task provides an opportunity for students to practice MAFS.K12.MP.2.1: Reason abstractly and quantitatively, and MAFS.K12.MP.3.1: Construct viable arguments and critique the reasoning of others.

Type: Problem-Solving Task

This task is designed to make students think about the meaning of the quantities presented in the context and choose which ones are appropriate for the two different constraints presented. In particular, note that the purpose of the task is to have students generate the constraint equations for each part (though the problem statements avoid using this particular terminology), and not to have students solve said equations. If desired, instructors could also use this task to touch on such solutions by finding and interpreting solutions to the system of equations created in parts (a) and (b).

Type: Problem-Solving Task

The primary purpose of this task is to lead students to a numerical and graphical understanding of the behavior of a rational function near a vertical asymptote, in terms of the expression defining the function.

Type: Problem-Solving Task

The purpose of this task is to give students practice constructing functions that represent a quantity of interest in a context, and then interpreting features of the function in the light of the context. It can be used as either an assessment or a teaching task.

Type: Problem-Solving Task

This problem involves the meaning of numbers found on labels. When the level of accuracy is not given we need to make assumptions based on how the information is reported. An unexpected surprise awaits in this case, however, as no reasonable interpretation of the level of accuracy makes sense of the information reported on the bottles in parts (b) and (c). Either a miscalculation has been made or the numbers have been rounded in a very odd way.

Type: Problem-Solving Task

## Tutorials

This video tutorial shows students: the standard form of a polynomial, how to identify polynomials, how to determine the degree of a polynomial, how to add and subtract polynomials, and how to represent the area of a shape as an addition or subtraction of polynomials.

Type: Tutorial

This video discusses multiplication and division of inequalities with negative numbers to solve the inequality.

Type: Tutorial

This tutorial demonstrates how to use the power of a power property with both numerals and variables.

Type: Tutorial

The video tutorial discusses about two typical polynomial multiplications. First, squaring a binomial and second, product of a sum and difference.

Type: Tutorial

This tutorial will help the learners practice multiplication of polynomials. Learners will understand that when they multiply expressions with more than two terms, they need to make sure each term in the first expression multiplies every term in the second expression.

Type: Tutorial

Binomials are the polynomials with two terms. This tutorial will help the students learn about the multiplication of binomials. In multiplication, we need to make sure that each term in the first set of parenthesis multiplies each term in the second set.

Type: Tutorial

In this tutorial, students will learn how to add and subtract polynomials functions using horizontal and vertical methods. In a horizontal format, like terms should be grouped together using the commutative property. In vertical format, terms should be listed by ascending degree with like terms placed below each other.

Type: Tutorial

In this tutorial students learn how to identify a polynomial, how to find the degree of a polynomial, and how to write a polynomial in standard format.

Type: Tutorial

This resource is a step-by-step tutorial on how to multiply polynomials.

Type: Tutorial

## Video/Audio/Animations

Khan Academy video tutorial on graphing linear equations: "Algebra: Graphing Lines 1"

Type: Video/Audio/Animation

This Khan Academy video tutorial introduces averages and algebra problems involving averages.

Type: Video/Audio/Animation

## Virtual Manipulatives

This resource will assess students' understanding of addition and subtraction of polynomials.

Type: Virtual Manipulative

This virtual manipulative is intended to allow the student to practice multiplication of binomials. The student should understand how to use algebra tiles before using this tool.

Type: Virtual Manipulative

This virtual manipulative allows the user to explore scenarios of a loan repayment by manipulating the amount of the loan, interest rate, payment amount, frequency of payments, and length of the loan in years.

Type: Virtual Manipulative

This virtual manipulative allows students to manipulate blocks, add or remove blocks, and connect them together to form solids. They can also experiment with counting the number of exposed faces, seeing what happens to the surface area when blocks are added or removed, and "unfolding" a block to create a net .

Type: Virtual Manipulative

This simulation allows the student to create a box plot and a histogram for the same set of data and toggle between the two displays. Maximum, minimum, median and mean are shown for the data set. The student can change the cell width to explore how the histogram is affected.

Type: Virtual Manipulative

Allows students access to a Cartesian Coordinate System where linear equations can be graphed and details of the line and the slope can be observed.

Type: Virtual Manipulative

This web address, from the National Library of Virtual Manipulatives, will help teachers and students validate the Pythagorean Theorem both geometrically and algebraically. It can be used interactively with the Smartboard and the Promethean Board to create a better understanding of the topic.

Type: Virtual Manipulative

This site provides a virtual balance on which the student can represent (and then solve) simple linear equations with integer answers. Conceptually, positive weights (unit-blocks and x-boxes) push the pans of the scale downward. Negative values are represented by balloons which can be attached to the pans of the scale. The student can then manipulate the weights to solve the equation while simultaneously seeing a visual display of these effects on the equation.

Type: Virtual Manipulative

Rotate shapes and their images with or without a background grid and axes.

Type: Virtual Manipulative

Section:Exceptional Student Education >Grade Group:Senior High and Adult >Subject:Academics - Subject Areas >